The locus of the foot of the perpendicular drawn from any focus to a variable tangent of the hyperbola $4x^2 - 9y^2 - 8x - 18y = 41$ is:

If $e_1$ and $e_2$ are the eccentricities of the hyperbola $16 x^2 - 9 y^2 = 1$ and its conjugate respectively,then $3 e_1 = $

If the angle between the asymptotes of the hyperbola $x^2-k y^2=3$ is $\frac{\pi}{3}$ and $e$ is its eccentricity,then the pole of the line $x+y-1=0$ with respect to this hyperbola is

Let the equation of two diameters of a circle $x^{2} + y^{2} - 2x + 2fy + 1 = 0$ be $2px - y = 1$ and $2x + py = 4p$. Then the slope $m \in (0, \infty)$ of the tangent to the hyperbola $3x^{2} - y^{2} = 3$ passing through the centre of the circle is equal to $......$

The foci of the hyperbola $4x^2 - 9y^2 - 36 = 0$ are:

Find the locus of the point of intersection of the lines $\sqrt{3}x - y - 4\sqrt{3}k = 0$ and $\sqrt{3}kx + yk - 4\sqrt{3} = 0$ for different values of $k$.